All automorphisms of the Calkin algebra are inner. (English) Zbl 1250.03094

Farah, Ilijas

Ann. Math. (2) 173, No. 2, 619-661 (2011).

Summary: We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a construction by N. C. Phillips and N. Weaver [Duke Math. J. 139, No. 1, 185–202 (2007; Zbl 1220.46040)] of an outer automorphism using the continuum hypothesis, this gives a complete solution to a problem of L. G. Brown, R. G. Douglas and P. A. Fillmore [Ann. Math. (2) 105, 265–324 (1977; Zbl 0376.46036)]. We also give a simpler and self-contained proof of the Phillips-Weaver result.

MSC:

03E35	Consistency and independence results
03E50	Continuum hypothesis and Martin’s axiom
46L05	General theory of \(C^*\)-algebras
46L40	Automorphisms of selfadjoint operator algebras

Keywords:

approximate homomorphisms; Calkin algebra; continuum hypothesis; independence results; outer automorphisms; Todorcevic’s axiom

Citations:

Zbl 1220.46040; Zbl 0376.46036

References:

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